From a vector calculus perspective, the CG coefficients associated with the SO(3)group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in quantum-mechanical terms can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and the integrals correspond to the Hilbert spaceinner product.[1] From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist complicated explicit formulas for their direct calculation.[2]

This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra so(3) ≅ su(2). This is physically interpreted as the square of the total angular momentum of the states on which the representation acts.

One can also define raising (j+) and lowering (j−) operators, the so-called ladder operators,

When two Hermitian operators commute, a common set of eigenfunctions exists. Conventionally j2 and jz are chosen. From the commutation relations the possible eigenvalues can be found. These states are denoted |jm⟩ where j is the angular momentum quantum number and m is the angular momentum projection onto the z-axis. They satisfy the following eigenvalue equations:

In principle, one may also introduce a (possibly complex) phase factor in the definition of C±(j,m){\displaystyle C_{\pm }(j,m)}. The choice made in this article is in agreement with the Condon–Shortley phase convention. The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized:

Here the italicized j and m denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman jx, jy, jz, j+, j−, and j2 denote operators. The δ{\displaystyle \delta } symbols are Kronecker deltas.

We now consider systems with two physically different angular momenta j1 and j2. Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or the orbital angular momenta of two electrons. These comprise two different irreducible representations of the same group, SO(3) (SU(2)), which combine together to another representation of the same group, but a reducible one. The reduction of this combined representation into a direct sum of irreducible ones comprises the Clebsch–Gordan series.

Let V1 be the (2 j1 + 1)-dimensional vector space spanned by the states

The Clebsch–Gordan coefficients ⟨j1m1j2m2 | JJ⟩ can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to the requirement that the norm of the state |[j1j2] JJ⟩ must be one.

The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with M = J − 1. Repeated use of that equation gives all coefficients.

This procedure to find the Clebsch–Gordan coefficients shows that they are all real in the Condon–Shortley phase convention.

Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore (−1)2k is not necessarily 1 for a given quantum number k unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule:

(−1)4k=1{\displaystyle (-1)^{4k}=1}

for any angular-momentum-like quantum number k.

Nonetheless, a combination of ji and mi is always an integer, so the stronger rule applies for these combinations:

(−1)2(ji−mi)=1{\displaystyle (-1)^{2(j_{i}-m_{i})}=1}

This identity also holds if the sign of either ji or mi or both is reversed.

It is useful to observe that any phase factor for a given (ji, mi) pair can be reduced to the canonical form:

(−1)aji+b(ji−mi){\displaystyle (-1)^{aj_{i}+b(j_{i}-m_{i})}}

where a ∈ {0, 1, 2, 3} and b ∈ {0, 1} (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this form is only locally canonical: it fails to take into account the rules that govern combinations of (ji, mi) pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of j1, j2, and j3 that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:

(−1)2(j1+j2+j3)=1{\displaystyle (-1)^{2(j_{1}+j_{2}+j_{3})}=1}

This identity also holds if the sign of any ji is reversed, or if any of them are substituted with an mi instead.

It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms a single spherical harmonic:

^The word "total" is often overloaded to mean several different things. In this article, "total angular momentum" refers to a generic sum of two angular momentum operators j1 and j2. It is not to be confused with the other common use of the term "total angular momentum" that refers specifically to the sum of orbital angular momentum and spin.